Summary: We provide a state-of-the-art survey of results concerning stability issues for routing in communication networks, and corresponding protocols. Roughly speaking, a distributed routing protocol is stable on some particular network if the protocol maintains a bounded number of packets in the network at all times, under some suitable assumptions on the fashion of injecting packets into the network. Fundamental questions that pose themselves in such a setting include: What is the effect of the injection pattern on stability? Which protocols are stable on which networks? What can be said about an algorithmic characterization of stability? In this survey, we summarize some of the known answers to these questions, and some questions for which no answers are yet known, as a collection of current challenges for the distributed computing community.